Transfer Maps and Orientation Classes of Singular Spaces

近年来，l理论、交同调和ad理论等拓扑方法，来自混合Hodge模理论的复代数方法，以及涉及迭代边黎曼度量和k同调的全局实解析方法的融合，导致了奇异空间的方向和特征类的密集组合的构建，这些方法通常是流形理论的经典不变量的推广。理解这些类在各种拓扑和代数Gysin限制下的行为，在光滑代数态射下的束传输映射和回拉开始成为可计算性和适用性的关键。因此，本项目旨在识别和建立在几个相关的同调理论，如边界、K-和l -同调的转移下奇异空间取向类的基本变换规律。预计这些结果将影响有关不变量相互关系的现有猜想。所发现的定律将为奇异空间的具体类型提供新的计算方法的基础，特别是超越理性同调流形的投影变体，例如舒伯特变体和数学物理中出现的其他变体。

The confluence of topological methods such as L-theory, intersection homology and ad-theories, complex algebraic methods coming from the theory of mixed Hodge modules, and global real analytic methods involving iterated edge Riemannian metrics and K-homology, has recently lead to the construction of a dense portfolio of orientation and characteristic classes for singular spaces that often generalize classical invariants of manifold theory. Understanding the behavior of these classes under various topological and algebraic Gysin restrictions, bundle transfer maps and pullback under smooth algebraic morphisms begins to emerge as a key to computability and applicability. The project thus aims for discerning and establishing the basic transformational laws governing orientation classes of singular spaces under transfers on several relevant homology theories such as bordism, K- and L-homology. The results are expected to impact existing conjectures concerning the relation of the invariants to each other. The laws found will then provide the foundation of new computational methods for concrete types of singular spaces, particularly projective varieties beyond rational homology manifolds, e.g. for Schubert varieties and other varieties arising in mathematical Physics.